Abstract

Long-range order and phase transitions in two-dimensional (2D) systems—such as magnetism, superconductivity, and crystallinity—have been important research topics for decades. The issue of 2D crystalline order has reemerged recently, with the development of exfoliated atomic crystals. Understanding the dimensional limit of crystalline phases, with different types of bonding and synthetic techniques, is at the foundation of low-dimensional materials design. We study ultrathin membranes of SrTiO3, an archetypal perovskite oxide with isotropic (3D) bonding. Atomically controlled membranes are released after synthesis by dissolving an underlying epitaxial layer. Although all unreleased films are initially single-crystalline, the SrTiO3 membrane lattice collapses below a critical thickness (5 unit cells). This crossover from algebraic to exponential decay of the crystalline coherence length is analogous to the 2D topological Berezinskii-Kosterlitz-Thouless (BKT) transition. The transition is likely driven by chemical bond breaking at the 2D layer-3D bulk interface, defining an effective dimensional phase boundary for coherent crystalline lattices.

INTRODUCTION

Arguments based on statistical mechanics (1) rigorously excluded the spontaneous breaking of continuous symmetries in two-dimensional (2D) systems, indicating the absence of long-range order (Mermin-Wagner theorem). However, it was later suggested that 2D systems can maintain a quasi-ordered phase at low temperature if they can host pairs of topological defects—such as vortex or dislocation pairs—which maintain continuous symmetry with finite energy (2–9). The unpairing of these topological defect pairs can collapse quasi–long-range order and introduce an unusual phase transition known as the Berezinskii-Kosterlitz-Thouless (BKT) transition, which was experimentally shown in a few systems (10–13). 2D crystalline order is one of the 2D systems initially predicted to undergo the BKT transition by having edge dislocation pairs as the topological defects (3, 5, 6, 9). However, the experimental study of the BKT transition in crystalline lattices has been rare, because the thermal fluctuation energy is typically hundreds of times smaller than chemical bonding energies in crystalline materials, limiting experimental studies to artificial 2D crystalline systems, such as colloidal spheres (14). Within this historical context, the discovery of graphene and its high-crystalline quality are extraordinary (15). It is now understood that graphene forms intrinsic ripples to stabilize the lattice without violating the Mermin-Wagner theorem (16–18). Although graphene and other atomic crystals represent examples of 2D crystalline phases without apparent phase transitions on accessible sample length scales, it is an open question to determine the general phase diagram for 2D lattices, with different chemical properties and bonding nature.

RESULTS

Structural characterization of ultrathin SrTiO3 membranes

SrTiO3 is a perovskite oxide compound with ionic bonding and has a cubic structure at room temperature. SrTiO3 is known for its wide physical and chemical stability windows; for example, the SrTiO3 (001) surface does not undergo reconstructions for moderate temperatures (19). The TiO2-terminated surface is stable in most common solvents; thus, SrTiO3 is a popular substrate for oxide thin film growth (20). The stability and the simple crystal structure suggest that SrTiO3 is an ideal crystalline lattice for studying 2D lattice physics without dominant contributions from chemical reactions and thermal decomposition. Here, we study the structural evolution of SrTiO3 in the 2D limit, taking advantage of recently developed methods for fabricating freestanding thin films (21, 22). SrTiO3 crystalline membranes of different thicknesses were synthesized by pulsed laser deposition (PLD), as previously reported for synthesis of large-area freestanding oxide layers (21). A water-soluble sacrificial layer (Sr3Al2O6) and the oxide film of interest (SrTiO3) were grown epitaxially on a SrTiO3 (001) substrate (Fig. 1A), and the sacrificial layer was selectively dissolved at room temperature in deionized water to release the top SrTiO3 layer. Both in situ and ex situ characterization confirmed that the oxide layers grown on the substrate were macroscopically single-crystalline, for which thicknesses were controlled with single u.c. (0.39 nm) accuracy (see Materials and Methods).

We examined the crystalline structure of freestanding SrTiO3 membranes (Fig. 1B) using TEM. Figure 1C shows HR-TEM images of SrTiO3 membranes of three different thicknesses (d). In the cases of relatively thicker membranes (8 u.c.), we found that the membrane is effectively single-crystalline with characteristic cubic lattice fringes. However, below a certain thickness, the film evolved to a mixture of crystalline and amorphous domains (4 and 3 u.c.) and further to an almost completely amorphous layer (2 u.c.). To quantify the observed crystalline phase transition, we used selected-area electron diffraction (SAED) patterns and dark-field TEM (DF-TEM) (23). SAED patterns of the entire suspended membrane (Fig. 2A) shows that the orientation of crystalline domains was mostly aligned within the 2-μm-wide area, even for the thinnest (2 u.c.) sample. DF-TEM, a diffraction-filtered imaging tool using the SAED pattern, was used for spatially mapping the crystalline domains in the membranes of different thicknesses (Fig. 2B).

(A) Diffraction patterns from suspended SrTiO3 membranes (signals from the entire suspended area of 2-μm diameter), from 6- (left), 4- (center), and 2-u.c.thick (right) membranes. The exposure time varies from 1 (6 u.c.) to 10 s (2 u.c.). The red circle indicates the size of the aperture for DF-TEM to selectively collect signals from crystalline domains. (B) DF-TEM images from membranes of different thicknesses, for which crystalline domains are of white color. Scale bars, 10 nm. (C) Thickness-dependent (d) crystalline coherence length (L) calculated from the spatial correlation function 〈G〉 deduced from DF-TEM images. Inset: The raw data of the correlation function from a 6-u.c. thick membrane.

There are two primary observations from the DF-TEM images. First, the membrane changed from approximately single-crystalline (white) to amorphous (black) as it got thinner, for which domain sizes were well matched to the HR-TEM images. Second, even for single-crystalline membranes, there was a thickness-dependent population of dislocations (local black areas). The density of dislocations observed in the membranes was much larger (~100 times) than the dislocation density originating from the substrate and thin film growth (21, 24, 25). We calculated spatial autocorrelation functions (26) for each thickness and estimated the crystalline coherence length (L) to quantify the crystalline order based on DF-TEM images (see Materials and Methods). Figure 2C shows the observed continuous reduction of the crystalline coherence length, with a qualitative change in trend near 5-u.c. thickness.

The observed crystalline to amorphous transition in the 2D limit is surprising, because all SrTiO3 layers were single-crystalline before release from the bulk substrates (Fig. 3). Several possible extrinsic factors to induce this transition—such as electron beam damage, surface chemical reactions, mechanical strain, and thermal effects (27)—were considered and examined in control experiments (see the Supplementary Materials). They show that SrTiO3 layers of a few u.c. thickness remain single-crystalline in all circumstances as long as the layer is epitaxially connected to a heterostructure on a bulk substrate, excluding dominant chemical reactions on the SrTiO3 surface. Instead, the crystalline phase transition occurs only during lift-off when the SrTiO3 membranes were detached, as confirmed by the surface crystalline structure observed by reflection high-energy electron diffraction (RHEED) in Fig. 3. The lift-off process involves bond breaking at the bottom interface via dissolution of the sacrificial layer (Sr3Al2O6), which releases a large amount of free energy before the broken bonds are passivated (28). This free energy, as large as a few electron volts for ionic bonds, appears to be the main factor driving the structural transition. By contrast, other fluctuation sources—such as relevant mechanical and thermal energies—are orders of magnitude smaller and thus negligible, consistent with the control experiments showing no effect of temperature up to 1150 K and several percent strains (figs. S4 and S5).

Fig. 3RHEED patterns of SrTiO3 films before and after solvent exposure and transfer.

Top: RHEED patterns of SrTiO3 films of different thicknesses grown on a Sr3Al2O6 20-nm film on a SrTiO3 (001) substrate. Middle: RHEED patterns of SrTiO3 films of different thicknesses grown on a Sr3Al2O6 5-nm film on a SrTiO3 (001) substrate after exposure to all solvents and polymer coatings used in the actual transfer process. The sacrificial layer (Sr3Al2O6) was too thin to be dissolved despite the long immersion in water; therefore, the SrTiO3 films are still epitaxially integrated to the 3D bulk crystalline heterostructure. Bottom: RHEED patterns of transferred SrTiO3 membranes (on Si/SiO2 substrates) of different thicknesses. The RHEED intensity of the 4-u.c. membrane diminished drastically and showed little signature of the crystalline phase. By contrast, the 8-u.c. membrane showed a similar RHEED pattern as the original film. No RHEED pattern was detected from the transferred 2-u.c. membrane.

BKT transition in the perovskite oxide membranes

Here, the structural transition shares much phenomenological similarity with the BKT transition expected for 2D lattices provided that the thickness can be understood to vary the effective “temperature” of the membrane. The BKT transition describes two phases separated by a critical temperature (TBKT) (3, 6, 7): a quasi-long-range–ordered topological phase and a disordered phase. At low temperature (T < TBKT), an effectively 2D crystalline lattice is stabilized with paired edge dislocations, the density of which increases algebraically with increasing temperature. At TBKT, where thermal fluctuations reach a fraction of the 2D dislocation energy (E2D), dislocations unbind and freely proliferate, resulting in the exponential decay of the crystalline coherence length. Both of these phases were observed in the SrTiO3 membranes of different thicknesses (Fig. 2B). Although the thermal fluctuation scale (kBT ~ 25 meV) is far below E2D even for the thinnest SrTiO3 membrane, electron volt–level fluctuation energy from bond breaking/passivation processes can be as large as E2D in membranes of a few u.c. thickness, driving a BKT-like transition (Fig. 4A). For a given fluctuation energy and Young’s modulus, thickness becomes the effective tuning parameter instead of temperature here, with the phase boundary defined by the critical thickness (dBKT).

(A) Phase diagram of crystalline order in 2D membranes. The phase boundary is set by the BKT transition energy (1/2E2D) proportional to the 2D modulus, thus the membrane thickness (see the Supplementary Materials) (5, 6). Thermal fluctuations at room temperature (kBT) are far below the BKT transition scale. However, fluctuation energy from interface bond breaking exceeds kBTBKT in the thin limit, collapsing the crystalline quasi-order for membranes thinner than the critical thickness (dBKT) during release from the bulk substrate. (B) Thickness-dependent dislocation area radius rd = (area)1/2 from DF-TEM images. Inset: An HR-TEM image of a dislocation for an 8-u.c. membrane. The green line encompassing the dislocation is closed, implying zero Burgers vector dislocation configuration. (C) A logarithmic plot of crystalline coherence length (L) versus (dBKT/d − 1)−ν for the membrane thicknesses below dBKT (2 to 5 u.c.). The gray line represents the least-square fitting of four data points assuming an exponential relation. Inset: critical exponent fitting from the crystalline coherence length, resulting in ν = −0.39.

On the basis of continuum elastic theory and the BKT transition model, we calculated the critical thickness (dBKT) for SrTiO3 membranes. A simplified atomic potential model was used to estimate E2D (deduced from the in-plane bonding potential E∥) and the chemical fluctuation energy (out-of-plane bonding potential E⊥) equivalent to kBTBKT. For an ionic atomic potential (Born-Landé model), dBKT can be expressed as(1)where α is the Born exponent of the lattice and a is the lattice constant (details are presented in the Supplementary Materials). In SrTiO3, where α is = 8 (29), for E⊥ = E∥ (isotropic bonding), dBKT is 6.3 u.c.

The snapshots of the evolution of crystalline coherence taken by DF-TEM (Fig. 2) enable quantitative analysis and provide compelling evidence for the transition based on BKT physics. We first examine the dislocation size in SrTiO3 membranes starting from the thick regime. Here, we found that nearly all dislocations are of zero Burgers vector configuration, as expected for paired dislocations (Fig. 4B, inset). The size was almost constant for 6-u.c. membranes and thicker membranes but increased drastically in 5- and 4-u.c. membranes (Fig. 4B). The size plot does not follow the percolation scaling law (30), which excludes the clustering of dislocations in a random manner. The abrupt size change from locally constrained dislocations (6 u.c.) to larger amorphous domains (5 u.c.) strongly suggests that there exists a critical thickness for a dislocation depairing transition between 5 and 6 u.c., close to the estimate from the atomic potential model (dBKT = 6.3 u.c. for isotropic bonding—note that the weak bonding anisotropy expected between SrTiO3 and Sr3Al2O6 would reduce this value). Another important indication of the BKT transition is the exponential scaling observed in the disordered phase. Figure 4C plots the crystalline coherence length as a function of the reduced thickness term of (dBKT/d − 1)−ν, analogous to the reduced temperature term t = (T/TBKT − 1)−ν in the original BKT theory. The critical exponent (ν) from the experimental data is −0.39 (Fig. 4C, inset), well matched to the theoretically predicted value (ν = −0.4) (3, 5, 6). Below the critical thickness, the coherence length is exponentially proportional to the reduced thickness term, which was also observed in other 2D phase transitions (11, 31). The exponential dependence is the characteristic scaling law of so-called infinite-order phase transitions, distinguishing the transition from second-order phase transitions with algebraic scaling.

General phase diagram of 2D crystalline lattices

The simple 2D phase transition model introduced here can be expanded to arbitrary crystalline lattices. Figure 5 describes the phase diagram of crystalline order for different thicknesses and bonding anisotropies (E∥/E⊥). The crystalline correlation length (ξ) is defined by the density of unpaired dislocations (5, 6, 9), identical to L in the disordered phase (d < dBKT) and infinite in the thicker membranes (d > dBKT). In materials with strong bonding anisotropy, interface bond breaking is a minimal energetic perturbation to the 2D crystal below the threshold to induce the phase transition. This corresponds to the case of atomic crystals exfoliated from layered compounds. Graphene and other layered materials have bonding anisotropy as large as several hundred, which excludes a visible crystalline phase transition in the exfoliated crystals and enables high crystalline quality down to the monolayer limit for all practical length scales (15, 32, 33). However, as E⊥ approaches E∥, interface bond breaking can provide sufficient fluctuation energy to drive the transition (Fig. 5). The most extreme case will be the isotropically bonded lattice. Although this phase diagram is based on a specific ionic bonding model, the general trend in the crystalline order versus bonding anisotropy will be similar for different types of chemical bonding.

Crystalline correlation length (ξ) as a function of the membrane thickness (d) and bonding anisotropy (E∥/E⊥) based on continuum elastic theory. Ionic bonding (Born-Lande model) lattice of α = 8 (corresponding to SrTiO3) and the numerical coefficients from Fig. 4C are assumed.

DISCUSSION

SrTiO3 membranes in the 2D limit present a unique experimental crystalline-amorphous system with a phase transition well described by BKT physics. This specific example also suggests the more general relevance of interface chemical energy for the stability of 2D crystalline order, providing a crucial guideline for the synthesis of 2D materials. Thus far, many 2D layers have been synthesized in a top-down approach by extraction from 3D crystals (15). Thus, interface bond breaking and corresponding fluctuations are unavoidable and should be considered as a critical component for determining the structural phase diagram. Unlike layered compounds, which naturally incorporate weak out-of-plane bonding, membranes derived from 3D crystalline lattices have a universal bound defined by the 2D-3D interface chemistry. This calls for new principles for designing chemical perturbations (for example, artificial interfaces and catalytic bond breaking). Understanding this intrinsic phase diagram of 2D crystalline order will open new opportunities for 2D materials of arbitrary crystalline structures, with controlled crystallinity.

MATERIALS AND METHODS

Sample preparation

SrTiO3/Sr3Al2O6 films were grown by PLD. On TiO2-terminated SrTiO3 (001) substrates, a 20-nm-thick Sr3Al2O6 film was grown in an Ar partial pressure of 5 × 10−6 torr with a laser fluence of 1.3 J/cm2, followed by SrTiO3 film growth at an O2 partial pressure of 5 × 10−6 torr with a laser fluence of 0.6 J/cm2. The substrate was preannealed at 950°C with an oxygen partial pressure of 5 × 10−6 torr for 30 min, and the growth temperature was maintained at 720°C for both Sr3Al2O6 and SrTiO3 layers. The entire film was grown in a layer-by-layer growth mode, for which the thickness was monitored by RHEED oscillations and later confirmed by atomic force microscopy (AFM) (fig. S1).

SrTiO3 membranes were transferred to TEM grids by the polymer-transfer method similar to the study of Huang et al. (fig. S2) (23). A thin layer (200 nm) of polymethyl methacrylate was spun onto the film and baked at 90°C, followed by additional application of polypropylene carbonate (>100 μm) as a mechanical support to prevent bending of membranes. The entire sample was immersed in deionized water at room temperature until the sacrificial layer of Sr3Al2O6 was completely dissolved, and the polymer-SrTiO3 layer was detached from the bulk substrate. The polymer-SrTiO3 layer was scooped by a silicon nitride TEM grid. The polymer support was removed by either thermal decomposition in a tube furnace heated at 250°C in flowing O2 or acetone wet etching followed by a critical point drying step. Both thermal decomposition and wet etching gave identical results for SrTiO3 membranes of the same thickness.

TEM image acquisition and image analysis

HR-TEM and DF-TEM experiments were conducted using FEI Tecnai G2 F20 X-TWIN TEM and FEI Titan 80-300 TEM, operated at either 80 or 300 kV. Acquisition times for DF-TEM images were 3 to 10 s. The spatial resolution of DF-TEM imaging is 1 nm. A Gatan 628-0500 in situ heating holder system was used to increase the sample temperature up to 900°C. All DF-TEM images were taken at the field of view where no image change was observed for more than 1-min exposure, preventing any change by the beam exposure during the measurement.

The angle-averaged spatial autocorrelation function 〈G(r)〉 of the DF-TEM images is the statistical correlation of two points separated by distance r and was calculated following the method of Giraldo-Gallo et al. (26). The characteristic length scale L from 〈G(r)〉 represents ξ in the disordered phase, although it approximately corresponds to the inverse square root of the paired dislocation density in the quasi-ordered crystalline phase. Each data point in Figs. 2 and 4 was calculated from the correlation functions of multiple (>10) images acquired from the membranes of the same thickness.

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Acknowledgments: We acknowledge L. Radzihovsky for valuable discussions and P. Giraldo-Gallo for technical support of data analysis. Funding: This work was supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract no. DE-AC02-76SF00515 (membrane synthesis and testing) and the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems Initiative through grant no. GBMF4415 (characterization of crystalline order). Part of this work was performed at the Stanford Nano Shared Facilities, supported by the NSF under award no. ECCS-1542152. Author contributions: S.S.H. and H.Y.H. designed the experiment. S.S.H. and D.L. synthesized and characterized the materials. J.H.Y., S.S.H., and A.F.M. carried out TEM measurements, supported by Y.C. and H.Y.H. S.S.H., Y.H., and H.Y.H. wrote the paper with input from all coauthors. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.